24,562 research outputs found
The Beylkin-Cramer Summation Rule and A New Fast Algorithm of Cosmic Statistics for Large Data Sets
Based on the Beylkin-Cramer summation rule, we introduce a new fast algorithm
that enable us to explore the high order statistics efficiently in large data
sets. Central to this technique is to make decomposition both of fields and
operators within the framework of multi-resolution analysis (MRA), and realize
theirs discrete representations. Accordingly, a homogenous point process could
be equivalently described by a operation of a Toeplitz matrix on a vector,
which is accomplished by making use of fast Fourier transformation. The
algorithm could be applied widely in the cosmic statistics to tackle large data
sets. Especially, we demonstrate this novel technique using the spherical,
cubic and cylinder counts in cells respectively. The numerical test shows that
the algorithm produces an excellent agreement with the expected results.
Moreover, the algorithm introduces naturally a sharp-filter, which is capable
of suppressing shot noise in weak signals. In the numerical procedures, the
algorithm is somewhat similar to particle-mesh (PM) methods in N-body
simulations. As scaled with , it is significantly faster than the
current particle-based methods, and its computational cost does not relies on
shape or size of sampling cells. In addition, based on this technique, we
propose further a simple fast scheme to compute the second statistics for
cosmic density fields and justify it using simulation samples. Hopefully, the
technique developed here allows us to make a comprehensive study of
non-Guassianity of the cosmic fields in high precision cosmology. A specific
implementation of the algorithm is publicly available upon request to the
author.Comment: 27 pages, 9 figures included. revised version, changes include (a)
adding a new fast algorithm for 2nd statistics (b) more numerical tests
including counts in asymmetric cells, the two-point correlation functions and
2nd variances (c) more discussions on technic
Simultaneous Coherent Structure Coloring facilitates interpretable clustering of scientific data by amplifying dissimilarity
The clustering of data into physically meaningful subsets often requires
assumptions regarding the number, size, or shape of the subgroups. Here, we
present a new method, simultaneous coherent structure coloring (sCSC), which
accomplishes the task of unsupervised clustering without a priori guidance
regarding the underlying structure of the data. sCSC performs a sequence of
binary splittings on the dataset such that the most dissimilar data points are
required to be in separate clusters. To achieve this, we obtain a set of
orthogonal coordinates along which dissimilarity in the dataset is maximized
from a generalized eigenvalue problem based on the pairwise dissimilarity
between the data points to be clustered. This sequence of bifurcations produces
a binary tree representation of the system, from which the number of clusters
in the data and their interrelationships naturally emerge. To illustrate the
effectiveness of the method in the absence of a priori assumptions, we apply it
to three exemplary problems in fluid dynamics. Then, we illustrate its capacity
for interpretability using a high-dimensional protein folding simulation
dataset. While we restrict our examples to dynamical physical systems in this
work, we anticipate straightforward translation to other fields where existing
analysis tools require ad hoc assumptions on the data structure, lack the
interpretability of the present method, or in which the underlying processes
are less accessible, such as genomics and neuroscience
Incremental Learning of Nonparametric Bayesian Mixture Models
Clustering is a fundamental task in many vision applications.
To date, most clustering algorithms work in a
batch setting and training examples must be gathered in a
large group before learning can begin. Here we explore
incremental clustering, in which data can arrive continuously.
We present a novel incremental model-based clustering
algorithm based on nonparametric Bayesian methods,
which we call Memory Bounded Variational Dirichlet
Process (MB-VDP). The number of clusters are determined
flexibly by the data and the approach can be used to automatically
discover object categories. The computational requirements
required to produce model updates are bounded
and do not grow with the amount of data processed. The
technique is well suited to very large datasets, and we show
that our approach outperforms existing online alternatives
for learning nonparametric Bayesian mixture models
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